Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Use a graphing utility to perform the operations for the matrices \(A, B,\) and \(C\) and the scalar \(c .\) Write a brief statement comparing the results of parts (a) and (b). $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 0 & -2 & 3 \\ 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & 0 \\ 4 & 1 & -2 \\ -1 & 2 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rrr} 3 & -2 & 1 \\ -4 & 0 & 3 \\ -1 & 3 & -2 \end{array}\right], \text { and } c=3 \end{aligned}$$. (a) \(A(B+C)\) (b) \(A B+A C\)

Step-by-Step Solution

Answer

The two matrices obtained from parts (a) and (b) are identical, hence, the operation of matrix multiplication is distributive over addition of matrices.

Step 1: Add matrices B and C

Add corresponding elements in the matrices \(B\) and \(C\) together. For example, the top left element of the new matrix will be the sum of the top left elements of \(B\) and \(C\), i.e. 2+3=5.

Step 2: Multiply matrix A with the sum of matrices B and C

Take each row in matrix \(A\) and multiply correspondingly with each column in the new matrix obtained in step 1, adding the products together to get the element in the same position in the result matrix. Do this for each element to complete the multiplication.

Step 3: Multiply matrix A with matrices B and C separately

Take each row in matrix \(A\) and multiply successively with each column in matrices \(B\) and \(C\), adding the products together to get the element in the corresponding position in the result matrices. Do this for all elements in both result matrices.

Step 4: Add the two result matrices together

Add corresponding elements in the two result matrices together to get the new matrix.

Step 5: Compare the result of (a) with (b)

Compare the two result matrices obtained from parts (a) and (b). If they are equal, then the operation of matrix multiplication is distributive over addition.

Step 6: Conclusion

Conclude whether the operation of matrix multiplication is distributive over addition based on the comparison result.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation, much like adding numbers. In matrix addition, you add corresponding elements from each matrix involved. For example, if you have two matrices, say matrix \(B\) and matrix \(C\), you will add each element in one matrix to the corresponding element in the other matrix.
To perform this operation, both matrices must be of the same dimensions. That means if matrix \(B\) is a 3x3 matrix, matrix \(C\) must also be 3x3 for you to add them. Here's how to do it in a simple way:

• Look at the positions in both matrices.
• Add the element in position (1,1) of matrix \(B\) to the element in position (1,1) of matrix \(C\), and so on.
• Continue this for all positions until you form a new matrix, which is the sum of \(B\) and \(C\).

This creates a new matrix where every elements represents the sum of the corresponding elements.
In our exercise, rows and columns are simply added one by one, giving us a new matrix before moving on with further operations.

Matrix Multiplication

Matrix multiplication is slightly more complex than matrix addition. Unlike simple arithmetic multiplication, you don’t multiply each element by its counterpart. Instead, matrix multiplication involves taking the rows of the first matrix and multiplying them against the columns of the second matrix.
To correctly multiply matrices, the number of columns in the first matrix must match the number of rows in the second matrix.

• First, take a row from the first matrix, in this case matrix \(A\).
• Multiply each element in this row by the elements in a column from the second matrix, say \(B\) or \(C\).
• Add all these products together to get a single number, which becomes an element in the result matrix.
• Perform this operation for all rows and columns to fill in the new matrix.

In our problem, the stepwise approach helps in illustrating how each row in \(A\) is multiplied with columns from the matrices obtained after addition, giving a resulting matrix from complex combined operations that follow matrix rules.
This results in new matrices for each operation, such as \(A(B+C)\) or \(A \times B\).

Distributive Property in Matrices

The distributive property is a familiar concept from arithmetic and algebra that also applies to matrices. It describes how matrix multiplication interacts with matrix addition. If you have matrices \(A\), \(B\), and \(C\), it states that the multiplication of a matrix through an addition can be distributed over each item in the addition like this:
\[ A \times (B + C) = A \times B + A \times C \]
This outlines the fact that multiplying a matrix by the sum of two matrices is equivalent to multiplying that matrix by each of the summands individually and then adding the results.
To check the distributive property in the provided exercise, you'd calculate \(A \times (B + C)\) and \(A \times B + A \times C\), then compare outcomes.

• First, add matrices \(B\) and \(C\).
• Then, perform multiplication with matrix \(A\) on the resulting sum.
• Separately, multiply \(A\) with \(B\) and \(A\) with \(C\), then add these two matrices.
• If the final matrices from these two methods match, the property holds true for the provided matrices.

This principle simplifies complex calculations, validating that distribution holds true even within matrix operations.